Let \( f(x) = x^4 + ax^3 + bx^2 + c \) be a polynomial with real coefficients such that \( f(1) = -9 \). Suppose that \( i\sqrt{3} \) is a root of the equation \( 4x^3 + 3ax^2 + 2bx = 0 \), where \( i = \sqrt{-1} \). If \( \alpha_1, \alpha_2, \alpha_3 \) and \( \alpha_4 \) are all the roots of the equation \( f(x) = 0 \), then \( |\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2 \) is equal to ________.
